Chapter 6.1, Problem 5E

(a)

To determine

To Sketch: The region R enclosed between the graph of the function y and x axis

Answer to Problem 5E

  

Explanation of Solution

Given information: The function $\begin{array}{l}y=2x-x^2\\ 0\le x\le 2\end{array}$

Sketch: We need to sketch the region bounded by function y and the x axis over the given interval.

Graph the function y by making a table. Assume some number for x and find out y

x	$y=2x-x^2$
-1	-3
0	0
1	1
2	0
3	-3

Graph the table on the coordinate plane

  

(b)

To determine

To Calculate: The area of the region R enclosed between the graph of the function y and x axis

Answer to Problem 5E

Approximate area of region R is 1.25

Explanation of Solution

Given information: The function $\begin{array}{l}y=2x-x^2\\ 0\le x\le 2\end{array}$

Calculation: Partition the given interval into 4 subintervals.

The width of each subinterval is

  $\begin{array}{l}\Delta x=\frac{b-a}{n}\\ a=0,b=2,n=4\\ \Delta x=\frac{4-2}{4}=0.5\end{array}$

  

Find the area of each rectangle by taking the left side value of each rectangle

  $\begin{array}{l}y=2x-x^2\\ x=0,y=0\\ x=0.5,y=0.75\\ x=1,y=1\\ x=1.5,y=0.75\end{array}$

The $y$ values are the height of the each rectangle. The width of each rectangle is 0.5

Area of the rectangle is width times height

Find the area of 4 rectangles and add the area

  $0.5·0+0.5·0.75+0.5·1+0.5·0.75=1.25$

Approximate area of region R is 1.25